Key Concepts for this section
1: Lorentz force law, Field, Maxwell’s equation
2: Ion Transport, Nernst-Planck equation
3: (Quasi)electrostatics, potential function,
4: Laplace’s equation, Uniqueness
5: Debye layer, electroneutrality
Goals of Part II:
(1) Understand when and why electromagnetic (E and B)
interaction is relevant (or not relevant) in biological
systems.
(2) Be able to analyze quasistatic electric fields in 2D
and 3D.
EM interactions in media - polarization (linear medium)
G
Eext = 0
G
G
E pol ∝ Eext
-+ -+ -+ -+
-
-+ -+ -+ -+
-
-+ -+ -+ -+
-
G
Eext
G
E pol
+
+
+
G
Eext
G
G
G
1 G
Emedia = Eext − E pol = Eext
εr
εr : relative permittivity (dielectric constant ) of the medium
(=>1)
ε of various media
Medium
εr
Water (pure)
~80
0.9% NaCl solution
~60
Ethanol
24
Methanol
34
Acetic acid
15~16
Gases
~1
Glass
3~4
Plastics and rubbers
2~9
Maxwell’s equation in a medium (Magnetic field)
Image source: MIT 8.02 class notes.
Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and Professor John W. Belcher. Used with permission.
G
G
Bmag ∝ Bext
G
G
G
G
Bmedia = Bext + Bmag = μr Bext ( μr ≥ 1)
G
G
G
G
G
∂E
∂E
∇ × B = μ 0 μ r J + μ 0 μ r ε 0ε r
= μ J + με
∂t
∂t
μr : relative magnetic permeability of the medium
μο : free space permeability (4π×10-7 H/m)
μ of various media
μr for water : very close to 1
μr (Ni)~600, μr (Fe)~5000
Mobility of various ions in water
Species
Mobility
Ui (cm2/v/s)
Diffusion coefficient
Di (cm2/s)
Cations in H2O (25oC)
H+
K+
Na+
Li+
36.30×10-4
7.62×10-4
5.19×10-4
4.01×10-4
9.33×10-5
1.96×10-5
1.33×10-5
1.03×10-5
Anions in H2O (25oC)
OHSO42ClNO3-
20.52×10-4
8.27×10-4
7.91×10-4
7.40×10-4
5.27×10-5
1.06×10-5
2.03×10-5
1.90×10-5
Electrons in Si at 25oC
Holes in Si at 25oC
1500
600
38.55
15.42
Comparative Number densities and Conductivities
Material
DI water
0.1M NaCl
Copper
Si (intrinsic)
Si (doped)
Nd=1016
Quartz
ni (#/cm3)
~1017
6×1019
~1022
n=p~1010
ne=1016
Np=104
σ (m-1Ω-1)
4 ×10-6
1.07
5.8 ×107
3.36 ×10-4
2.4
10-18
In silicon (semiconductor), n×p~1020 (constant)
In aqueous solutions, [H+][OH-] = 10-14 = Kw
(pH= -log10[H+])
Electrolytes (biological systems) are conductors.
electrode
E
V=Vext
Charge Relaxation in electrolyte
+
+
+
++ ++
+++
+
τ
+
+
~10-9 sec
+
ρ=0
+
+
<ρ>=0 within the medium
Fixed charges
+
+
+
+
τ ~10-9 sec
- - --- +
- -
+
+
+
+
+
<ρ>=0 within the medium
• (Quasi) Electroneutrality Approximation
– It is an approximation.
• Valid only after the relaxation time constant τ,
and outside of the Debye length (κ-1, to be
discussed)
– Not valid when there is a discontinuity
• Valid only within a single medium (σ=constant)
• Boundary of the medium could carry (surface)
free charge
– No inter-charge interaction in liquid media
Electroneutrality
Buffer counterions
~ Debye length (κ-1)
+ --
- -
Protein A
+
+
+ +
+
Protein B
Capillary Electrophoresis (1980s)
Electroosmotic flow
(UV; laser
fluorescence
and mass spec; radioisotope)
Capillary inlet
Data acquisition
Capillary outlet
10 - 200 µm
Detector
Electrolyte buffer
Electrolyte buffer
+
Reservoir
105 V/m
+
Reservoir
HV
Generic diagram of a capillary electrophoresis system.
Figure by MIT OCW.
Micro Total Analysis System (microTAS): Parallelism
• 96~356 samples analyzed in a single chip simultaneously
• fluorescence detection of DNA at the center of the chip (rotating
optical head)
Figure 1 removed due to copyright restrictions.
Yining Shi et al., Analytical Chemistry, 71, 5354 (1999)